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Anisotropic Bragg diffraction of finite-sized volume holographic grating in photorefractive crystals
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Optik
Optics
Optik 118 (2007) 418–424 www.elsevier.de/ijleo
Anisotropic Bragg diffraction of finite-sized volume holographic grating in photorefractive crystals Aimin Yana,, Lingyu Wanb, De’an Liua, Liren Liua a Information Optics Laboratory, Shanghai Institute of Optics and Fine Mechanics, The Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, China b College of Physics Science and Technology, Guangxi University, Nanning 530004, China
Received 16 January 2006; accepted 10 March 2006
Abstract Anisotropic diffraction of uniform plane wave by finite-sized volume holographic grating in photorefractive crystals is considered. It is found that the anisotropic diffraction can take place when some special conditions are satisfied. The diffracted image is obtained in experiment for the anisotropic Bragg diffraction in Fe:LiNbO3 crystals. A coupled wave analysis is presented to study the properties of anisotropic diffraction. An analytical integral solution for the amplitudes of the diffracted beams is submitted. A trade off between high diffraction efficiency and the deterioration of reconstruction fidelity is analyzed. Numerical evaluations also show that the finite-sized anisotropic volume grating exhibits strong angular and wavelength selectivity. All the results are useful for the optimizing design of VHOE based on finite-sized volume grating structures. r 2006 Elsevier GmbH. All rights reserved. Keywords: Diffraction optics; Finite-sized volume holographic grating; Coupled-wave theory; Anisotropic diffraction
1. Introduction Holographic recording of anisotropic photorefractive grating in electro-optic lithium niobate crystals (LiNbO3) [1,2] has been extensively studied during recent decades for the purpose of recording highly efficient volume phase holograms. Anisotropic diffraction, due to the polarization orthogonality of the diffracted and undiffracted beams and improved signal-to-noise ratio, has been widely used in signal processing and holographic optical elements such as polarizers, switches and connectors. For the devices to be small, the two-dimensional (2-D) finite-sized volume Corresponding author. Fax: 0086 21 6991 8096.
E-mail address: [email protected] (A. Yan). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.03.034
grating [3] (or called as crossed-beam volume grating) is often encountered in many practical applications. The size of the volume grating in the 2-D plane in which the grating vector lies is restricted. They can easily form right-angle diffraction grating in waveguide devices and support the most space-efficient device designs [4,5]. As we all know, the most commonly used theory for diffraction by volume holograms is the coupled-wave theory of Kogelnik [6]. It can be also applied to deal with the problem of anisotropic diffraction [1,2,7,8] in photorefractive crystals. Recently, Montemezzani and Zgonik [9] modified the coupled-wave theory of Kogelnik to include the effects of mixed phase and absorption for grating vector and medium oriented in arbitrary directions to the case of anisotropic materials. The resultant widely used simple analytical expressions
ARTICLE IN PRESS A. Yan et al. / Optik 118 (2007) 418–424
2. Experiment setup and results A schematic diagram of experimental arrangement for observing the anisotropic Bragg diffraction is shown in Fig. 1. The sample used in our experiments is a single domain (0.15 wt% Fe2O3) and oxidized Fe: LiNbO3 crystals with 4 4 10 mm3. A He–Ne laser beam (l ¼ 632:8 nm), which is polarized as an extraordinary light using a half-wave plate is split into two beams with nearly equal intensity 10.5 mW. We expand the two beam sizes, whose diameters are nearly both 4 mm. The two beams are incident upon the two different surface of
BE
M
M BS
1 λ 2
C-axis(z)
.
He-Ne Laser
for isotropic [6] and anisotropic [9] gratings have been found to yield extremely good results when the corresponding assumptions are satisfied. However, these analyses are for the diffraction of infinite and planar boundary volume gratings rather than the practical case of finite-sized gratings. Previous studies [10–12] of the diffraction for finite-sized volume holographic gratings are mostly discussed their isotropic diffraction properties. Solymar [13] has treated such volume gratings with a fully 2-D coupled-wave theory. And analytical solutions with an elegant form have been obtained for certain geometrical configurations. But in this theory, the polarization states of incident and diffracted beam are identical; no investigations of anisotropic diffraction are treated. Generally, with the rapid development of the minimization and integration of optical elements in integrated optics and optical communication, one frequently encounters anisotropic diffraction problem with the transformation of polarization states for finite-sized volume gratings. Therefore, a knowledge of the anisotropic diffraction characteristics of such gratings would be very valuable in practical applications. The present investigation concentrates on the problem of anisotropic diffraction for the most interesting case of finite-sized volume holographic gratings recorded in photorefractive crystal. As for the specific recorded isotropic holograms, when the anisotropic Bragg condition is satisfied, the readout beam can make the anisotropic diffraction beam appear. Corresponding experimental results are presented in Section 2. The detailed phase-matching condition of anisotropic diffraction is discussed in the following Section 3. In Section 4, consistent with the 2-D coupled-wave theory, an analytical solution of the diffraction beam has been obtained. Furthermore, we present a detailed theoretical analysis of the several diffraction properties such as angular and wavelength selectivity, trade off between high efficiency and reconstruction fidelity of finite-sized photorefractive volume holographic gratings. A conclusion is given in Section 5.
419
SH
Crystal CCD
M
Fig. 1. Experimental setup: M, rotatable mirror; 1/2l, halfwave plate; BE, beam expander; BS, beam splitter; SH, shutter.
Fig. 2. Experimental results: the photos of diffracted images of (a) isotropic and (b) anisotropic diffraction observed on a screen behind the crystal.
4 mm 4 mm plane with the intersection angle of 2y0 r outside crystal (y0r ¼ 451). Optical axis c is perpendicular to the plane containing the grating vector and the wave vector of the readout beam. We assume that the propagation and polarization directions of the incident beam are in the (x,y) plane, i.e., the ordinary mode. And the wave polarized along z direction is defined as extraordinary mode. A reflection mirror mounted on a fine rotating stage is used in readout phase for rotating the incidence angle of the readout beam. A CCD detector with viewing screen is placed behind the crystal and is parallel to the output surface of the diffracted beams. We used the two beam interference scheme to record the grating that is a similarity of usually used in holographic experiments. The only difference is the two recording beams entering two different crystal surfaces. In the experiment, the intersection angle inside the crystal is equal to 901 because the two recording beams are incident normal to the crystal surfaces, respectively. After the recording, one of the extraordinary beams is blocked and the other extraordinary beam is diffracted from the written grating. Therefore, the isotropic diffraction (or called as EE diffraction) can take place because of satisfying automatically the
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isotropic Bragg condition. A diffracted image can be observed on the screen and is shown as in Fig. 2(a). Then we changed the readout angle at about the Bragg readout angle y0 i by rotating the reflection mirror. That was chosen to read the recorded grating with the angle y0 i to satisfy the Bragg matching condition of anisotropic diffraction which will be discussed in next section in detail. We can observe the anisotropic diffraction (or called as extraordinary to ordinary (EO) diffraction) at the Bragg diffraction angle y0 d using the CCD detector. Photograph of the anisotropic diffracted image is shown in Fig. 2(b).
where e0 and De are the unperturbed and perturbed permittivity tensor of the photorefractive crystal, no and ne are the ordinary and extraordinary indices, respectively; the reduced subscript notation rij represents rijk in accordance with the widely used convention. Now, for the symmetrical recording geometry of the photoinduced refractive index grating as shown in Fig. 3, the grating vector Kg is along y axis and the two off-diagonal elements n2o n2e r42 E sc are nonzero. y Therefore, in this case the EO diffraction of an extraordinary wave into an ordinary wave and vice versa is possible. The phase-matching condition required for observing anisotropic diffraction is
3. Anisotropic Bragg matching condition in LiNbO3 crystal
K g ¼ k i kd
Diffraction of volume grating is a process of wave coupling. Generally, the generation of anisotropic Bragg diffraction needs two necessary conditions, i.e., the interacting beams are phase matched and the dielectric tensor elements involved are nonzero. Photorefractive ferroelectric LiNbO3 is a typical electro-optic crystal. In crystal inhomogeneous illumination excites electrons from defects into the conduction band. These free charge carriers migrate and trapped in darker regions, and space-charge fields build up. For simplicity, we assume that the two recording beams are both extraordinary polarization and the space-charge field is steady-state and written as E ¼ E 0 ½expðjK g rÞ þ expðjK g rÞK^ g , sc
where E0 is the amplitude of the space-charge field, K^ g ¼ K g =jK g j is unit grating vector. This space-charge field modulates the index of refraction through the linear electro-optic effect [14], 1 D ¼ rjkl E sc (1) l , jk where e is the relative permittivity tensor, D(1/e)jk is the change in the jkth element of 1/e, and rjk is the electrooptic tensor. For a LiNbO3 crystal (of 3 m point group) of which the optical axis is parallel to z axis, the permittivity tensor that is subject to the electro-optic effect with the photoinduced space charge fieldE sc j ¼ sc sc ½E sc ; E ; E , can be expressed as, x y z
2
n2o 6 ¼ 0 þ D ¼ 4 0 0
0 n2o 0
(3)
and also illustrated by the readout wave-vector diagram shown in Fig. 4. In other words, the triangle consisting of incident light wave vector ki, diffracted light wave vector kd and grating vector Kg should be complete. Based on momentum conservation principle, we can write the Bragg condition for the anisotropic diffraction in (x, y) plane of uniaxial crystal [15] as following expressions, ne sin yi þ no sin yd ¼ 2ne sin yr , ne cos yi ¼ no cos yd ,
ð4Þ
sinð45 y0i Þ ¼ ne sinð45 yi Þ, sinð45 y0d Þ ¼ no sinð45 yd Þ, sinð45 y0r Þ ¼ ne sinð45 yr Þ,
ð5Þ
where yr, yi and yd are the internal writing angles of the two incident beams and one diffracted beam with respect to the x-axis, y0 r, y0 i and y0 d are the corresponding external writing angles and diffraction angle, respectively. For a grating written with 632.8 nm at an incident angle of 451, Eqs. (4) and (5) theoretically predict a Bragg angle of 40.071 and 50.511 for incidence and diffraction y0 i and y0 d of anisotropic diffraction. (principal refractive indices of LiNbO3 crystal at l ¼ 632:8 nm are: no ¼ 2:286, ne ¼ 2:200)[16]. So the variation of the readout angles for isotropic and anisotropic diffraction in theory can be calculated as dy0 ¼ 4:931.
3 3 2 n4 ðr E sc r E sc Þ n4o r22 E sc n2o n2e r42 E sc 13 z 0 o 22 y x x 6 sc sc 7 2 2 n4o r22 E sc n2o ðr22 E sc 07 5þ6 x y þ r13 E z Þ no ne r42 E y 7 4 5 sc sc 2 2 4 n2e n2o n2e r42 E sc n n r E n r E 42 33 o e e x y z
(2)
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421
y y Grating boundary crystal
r
0
Kg x
i
r 45°
d
R
S
i
WS
i
d 45°
x
d
WR (a)
(b)
Fig. 3. (a) Geometry for the recording and anisotropic diffraction of the finite-sized volume grating and (b) the refractive at crystal surface.
denoted as ki, kd and polarization unit vectors ei, ed. Here ki ¼ ki ðx cos yi þ y sin yi Þ, ki ¼ 2pne =l. From Eq. (3), we can obtain the wave vector of diffracted beam kd ¼ kd ðx cos yd y sin yd Þ, where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kd ¼ k2i þ K 2g 2ki K g sin yi , sin yd ¼ ðK g ki sin yi Þ
x
Kg
ki
i
d
=kd and cos yd ¼ ðki cos yi Þ=kd . To determine the wave coupling, the total optical field Eq. (6) must also approximately satisfy the wave equation:
kd
y
r2 EðrÞ þ o2 m0 ½0 þ DEðrÞ ¼ 0,
Fig. 4. Wave-vector diagram illustrating how anisotropic diffraction is produced via a finite-sized grating of wave vector Kg in LiNbO3 crystal (optical axis c is perpendicular to the incidence plane).
4. Anisotropic diffraction in finite-sized volume grating We shall investigate in this section the anisotropic Bragg diffraction of a finite-sized photorefractive volume grating recorded by two finite plane waves of uniform amplitude distribution. When such a monochromatic optical beam, entering the photorefractive crystal at or near the Bragg angle, is diffracted by an existing photoinduced refractive index grating, only two significant light beams are assumed to be present, namely, the diffracted and transmitted beams. The total optical field in the grating may be written in phasor form as EðrÞ ¼ ei R0 ðrÞ expðjki rÞ þ ed S0 ðrÞ expðjkd rÞ,
where o is the angular frequency of the light inside the photorefractive crystal. For the EO type anisotropic diffraction mode (the propagation and polarization directions of the diffracted wave are in the xy plane and the incident wave polarized along z direction), we can rewrite the total optical field vector as 2 3 edx S0 ðx; yÞ exp½jkd ðx cos yd y sin yd Þ 6 7 EðrÞ ¼ 4 edy S 0 ðx; yÞ exp½jkd ðx cos yd y sin yd Þ 5. eiz R0 ðx; yÞ exp½jki ðx cos yi þ y sin yi Þ (8) Substituting Eqs. (2) and (8) into the wave equation, with the slowly varying envelope assumption (SVEA) and after some algebra, using the relations e2dx þ e2dy ¼ 1 and eiz ¼ 1, we can derive the following coupled-wave equations for the readout and diffracted beams, cos yi
qR0 qR0 þ sin yi ¼ jkR S 0 , qx qy
(9a)
cos yd
qS 0 qS 0 sin yd þ jWS0 ¼ jkS R0 , qx qy
(9b)
(6)
where R0(r) and S0(r) are the complex amplitudes of the readout and diffracted beams, with propagation vectors
(7)
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where kR ¼ ðo2 =2c2 ki Þn2o n2e r42 E 0 ðed K^ g Þ, kS ¼ ðo2 =2c2 kd Þn2o n2e r42 E 0 ðed K^ g Þ and W denotes the deviations form the Bragg condition W ¼ ððo2 =c2 Þn2o k2d Þ=2kd . As shown by Parry and Solymar [17], introducing a new coordinate system (l, m) defined by l ¼ x sin yi y cos yi and m ¼ x sin yd þ y cos yd results in a simplified version of the coupled wave equations, with R0(x, y) and S0(x, y) being transformed into R(x, y) and S(x, y). The directions of the coordinates l and m are at right angles to the directions of the reference and signal beams in readout phase, respectively. The new equations are q2 S þ k0R k0S S ¼ 0 qlqm
and
1 qS , R¼ 0 jkS ql
(10)
where Rðl; mÞ ¼ R0 ðl; mÞ expðjW0 lÞ, Sðl; mÞ ¼ S0 ðl; mÞ expðjW0 lÞ, k0R ¼ kR = sinðyi þ yd Þ, k0S ¼ kS = sinðyi þ yd Þ and W0 ¼ W= sinðyi þ yd Þ. The boundary conditions may be expressed in the l and m coordinate system as S 0 ðl; f ðlÞÞ ¼ 0 on m ¼ f ðlÞ,
(11)
qS 0 ¼ jk0S expðjW0 lÞ on m ¼ qðlÞ, (12) ql where f ðlÞ ¼ sinðyd þ yr Þ= sinðyi yr Þl and qðlÞ ¼ sinðyd yr Þ= sinðyi þ yr Þl. From Fig. 3, we can see that the output boundaries of the transmitted and diffracted beams are parallel to the input boundaries, respectively. But the curves of the output boundaries are very complex and not easy to express the amplitude distributions of the two output beams. So we introduce another coordinate system u ¼ x sin yr y cos yr and v ¼ x sin yr þ y cos yr . In the other new coordinate system, the boundary conditions of output beams can be simplified as u ¼ W R and v ¼ W S . The relationship between (l, m) and (u, v) coordinate systems is written as l ¼ gðu; vÞ ¼ sinðyi þ yr Þu þ sinðyi yr Þv= sin 2yr , u sinðyd yr Þ þ v sinðyd þ yr Þ m ¼ hðu; vÞ ¼ . sin 2yr
The diffraction efficiency [19] is defined as DE ¼ Pd =Pi , 1
R
(16) 0
ÞS 0 ðW R ; v0 Þ dv0
and Pi ¼ where S0 ðW R ; v R Pd ¼ Z Z 1 R0 ðu0 ; 0ÞR0 ðu0 ; 0Þ du0 are respectively the total energy of the diffracted beam and incident readout beam, Z is the average characteristic impedance of the medium.
5. Numerical results and discussions In this section, we discuss the diffraction properties of the finite-sized volume grating utilizing solutions Eq. (15) of coupled wave equations. Fig. 5 reveals the evolution of the coupling process for various values of space charge field amplitude E0 (0.04, 0.1, 0.8 V/mm,) for an on-Bragg anisotropic diffraction, the corresponding diffraction efficiency is indicated in parentheses. As can be expected, the amplitude of diffracted beam and the diffraction efficiency increase with the value of E0 because of the increasing value of the refractive index modulation depth and the coupling constant. In some practical applications, a high efficiency of finite-sized volume hologram is of course needed. On the other hand, from Fig. 5 we can show that higher diffraction efficiencies imply gross distortion of the reconstructed amplitude distribution for the discussed uniform modulation grating. The recording and readout beams we assumed are plane, the amplitude distributions uniform and have a finite lateral width. But the diffraction amplitude deviates more and more from the uniform amplitude distribution. It may be concluded that there is a trade-off between high efficiency and reconstructed uniform beam-profile. This characteristic is very similar to that of isotropic diffraction finite-sized volume grating. If we want to obtain simultaneously with both
(13)
2.0 E0=0.8 V/µm (62%) E0=0.1 V/µm (39%)
(14)
E0=0.04 V/µm (6%)
1.5
The solution for the diffracted wave of the secondorder hyperbolic partial differential equation with its corresponding boundary conditions may be obtained by means of the so-called Riemann method [18]. Then we have Z lA S 0 ðwR ; vÞ ¼ jk0S exp½jW0 gðW R ; vÞ expðiW0 l 0 Þ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 0 ð2 k0R k0S ½l 0 gðW R ; vÞ½f ðl 0 Þ hðW R ; vÞÞ dl 0 ,
ð15Þ where J0 and J1 are, respectively Bessel functions of the zeroth and first order, l A ¼ W R sinðyi þ yr Þ= sin 2yr .
|S|
422
1.0
0.5
0.0
0
1
2 v (mm)
3
4
Fig. 5. Amplitude profiles of diffracted beam for several values of space charge field E0, using W R ¼ W S ¼ 4 mm.
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high diffraction efficiency and good reconstruction fidelity, some special techniques for optimizing beam uniformity such as quasi-plane wave illumination, nonuniform grating strength modulation, ideal boundary shape, et al. must be used. This phenomenon can also be of great practical importance in the optimizing design of a VHOE device. In many practical cases, it is of interest to investigate the anisotropic diffraction behavior at or near the Bragg matching condition. To fulfill the above condition, we introduce the Bragg mismatching parameter in coupled wave equation Eq. (9b) and which is defined by 2 o 2 pðn2o n2e Þ ne 2 W¼ þ Kg n k 2kd ¼ d o 2 no l c no sin yi
K 2g 4pno
l.
ð17Þ
In this section, we assume that when the Bragg condition Eq. (3) is satisfied, the Bragg incident angle and corresponding wavelength are, respectively denoted as yi0 and l0. The parameters of Dy ¼ yi yi0 and Dl ¼ l l0 are the angle and wavelength deviations from the Bragg condition, respectively. And the deviations Dy and Dl are small. Therefore, a first-order Taylor series expansion of Eq. (16) yields the following expression: " # K 2g ne pðn2o n2e Þ Dl W ¼ K g cos yi0 Dy Dl þ . no l0 l0 no 4pno (18) Note Eq. (18) differs from the corresponding expression for W in Ref. [6]. Figs. 6(a and b) show the dependence of the diffraction efficiency DE on the angular and wavelength mismatch parameters Dy and Dl (W R ¼ W S ¼ 4 mm, l0 ¼ 633 nm, y0 ¼ 42:761, E 0 ¼ 0:04 V=mm). It is demonstrated clearly that the diffraction efficiency of anisotropic diffraction decreases dramatically for small values of Dy in Fig. 6(a). Its maximum value concentrates at the center coordinate, whereas Dy is zero. The same situation also can be found for Dl in Fig. 6(b). These effects are due to the loss of coupling between the two beams inside the volume grating when off-Bragg parameters are introduced. It can be concluded that finite-sized anisotropic volume gratings exhibit strong angular and wavelength selectivity. The phenomena can find applications in the fields of optoelectronics such as optical data storage, angular and wavelength filters in DWDM systems.
6. Conclusions The anisotropic Bragg diffraction of light by finitesized volume grating in photorefractive crystals is
Fig. 6. (a) The variation curve of diffraction efficiency as a function of angular deviation Dy, Dl ¼ 0. (b) The variation curve of diffraction efficiency as a function of wavelength deviation Dl, Dy ¼ 0.
investigated in this paper. This diffraction can take place when the Bragg phase matching condition is satisfied and the involved off-diagonal tensor elements are nonzero. We also have identified the anisotropic diffraction behavior in experiment for the volume grating recorded in Fe:LiNbO3 crystal; the photos of diffracted images of isotropic and anisotropic diffraction are given. The theoretical model for the anisotropic diffraction of finite-sized volume grating is presented at or near Bragg diffraction. The model is based on the two-wave first-order coupled wave theory framework developed by Solymar. Analytical solutions are derived for amplitudes of the transmitted and diffracted beams. Numerical calculations show that, for EO-type Bragg diffraction of uniform plane waves, high reconstruction fidelity is often accompanied by a decrease in diffraction efficiency. The phenomenon can be of an important problem considered in practical design of VHOE. For deviations from the anisotropic Bragg condition, it is
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shown that the diffraction efficiency depends strongly on the deviation parameter Dy and Dl. In particularly, the finite-sized anisotropic volume grating exhibits high angular and wavelength selectivity. All the results are useful for the design of volume holographic optical elements based on finite-sized anisotropic volume grating.
Acknowledgments The authors acknowledge the supports of the Science and Technology Committee of China (Grant No. 2002CCA03500) and the National Nature Science Foundation of China (Grant No. 60177016).
References [1] D.A. Temple, Warde, Anisotropic scattering in photorefractive crystals, J. Opt. Soc. Am. B 3 (1986) 337–341. [2] M.P. Petrov, T.G. Pencheva, S.I. Stepanov, Light diffraction from volume phase holograms in electrooptic photorefractive crystals, J. Opt. (Paris) 12 (1981) 287–292. [3] P.S.T.J. Russell, L. Solymar, The properties of holographic overlap gratings, Opt. Acta 26 (1979) 329–347. [4] M.L. Jones, R.P. Kenan, C.M. Verber, Rectangular characteristic gratings for waveguide input and output coupling, Appl. Opt. 34 (1995) 4149–4158. [5] G. Burr, F. Mok, D. Psaltis, Angle and space multiplexed holographic storage using the 90-degree geometry, Opt. Commun. 117 (1995) 49–55. [6] H. Kogelnik, Coupled wave theory for thick hologram gratings, Bell Syst. Tech. J. 48 (1969) 2909–2947.
[7] J.J. Butler, M.S. Malcuit, M.A. Rodriguez, Diffractive properties of highly birefringent volume gratings: investigation, J. Opt. Soc. Am. B 19 (2002) 183–189. [8] A.M. Lopez, M.P. Arroyo, M. Quintanilla, Some polarization effects in holographic volume gratings, J. Opt. A: Pure Appl. Opt. 1 (1999) 378–385. [9] G. Montermezzani, M. Zgonik, Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries, Phys. Rev. E 55 (1997) 1035–1047. [10] M.G. Moharam, T.K. Gaylord, R. Magnusson, Diffraction characteristics of three-dimensional crossed-beam volume gratings, J. Opt. Soc. Am. 70 (1980) 437–442. [11] G. Notni, R. Kowarschik, Diffraction analysis of threedimensional volume gratings with arbitrary boundaries, J. Opt. Soc. Am. A 6 (1989) 1682–1691. [12] L. Solymar, M. Jordan, Two-dimensional transmission type volume holograms for incident plane waves of arbitrary amplitude distribution, Opt. Quantum Electron. 9 (1977) 437–444. [13] L. Solymar, A general two-dimensional theory for volume holograms, Appl. Phys. Lett. 31 (1977) 820–822. [14] W.J. Burke, D.L. Staebler, W. Phillips, G.A. Alphonese, Volume phase holographic storage in ferroelectric crystals, Opt. Eng. 17 (1978) 308–316. [15] R.W. Dixon, Acoustic diffraction of light in anisotropic media, IEEE J. Quantum Electron. QE-3 (1967) 85–93. [16] A. Yariv, P. Yeh, Optical Waves in Crystals, Wiley, New York, 1984, pp. 121–154. [17] W.E. Parry, L. Solymar, A general solution for twodimensional volume gratings, Opt. Quantum Electron. 9 (1977) 527–531. [18] R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. II, Interscience, New York, 1962 Chapter V. [19] S. Tao, B. Wang, G.W. Burr, J. Chen, Diffraction efficiency of volume grating with finite size: corrected analytical solution, J. Mod. Opt. 51 (2004) 1115–1122.
Optik
Optics
Optik 118 (2007) 418–424 www.elsevier.de/ijleo
Anisotropic Bragg diffraction of finite-sized volume holographic grating in photorefractive crystals Aimin Yana,, Lingyu Wanb, De’an Liua, Liren Liua a Information Optics Laboratory, Shanghai Institute of Optics and Fine Mechanics, The Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, China b College of Physics Science and Technology, Guangxi University, Nanning 530004, China
Received 16 January 2006; accepted 10 March 2006
Abstract Anisotropic diffraction of uniform plane wave by finite-sized volume holographic grating in photorefractive crystals is considered. It is found that the anisotropic diffraction can take place when some special conditions are satisfied. The diffracted image is obtained in experiment for the anisotropic Bragg diffraction in Fe:LiNbO3 crystals. A coupled wave analysis is presented to study the properties of anisotropic diffraction. An analytical integral solution for the amplitudes of the diffracted beams is submitted. A trade off between high diffraction efficiency and the deterioration of reconstruction fidelity is analyzed. Numerical evaluations also show that the finite-sized anisotropic volume grating exhibits strong angular and wavelength selectivity. All the results are useful for the optimizing design of VHOE based on finite-sized volume grating structures. r 2006 Elsevier GmbH. All rights reserved. Keywords: Diffraction optics; Finite-sized volume holographic grating; Coupled-wave theory; Anisotropic diffraction
1. Introduction Holographic recording of anisotropic photorefractive grating in electro-optic lithium niobate crystals (LiNbO3) [1,2] has been extensively studied during recent decades for the purpose of recording highly efficient volume phase holograms. Anisotropic diffraction, due to the polarization orthogonality of the diffracted and undiffracted beams and improved signal-to-noise ratio, has been widely used in signal processing and holographic optical elements such as polarizers, switches and connectors. For the devices to be small, the two-dimensional (2-D) finite-sized volume Corresponding author. Fax: 0086 21 6991 8096.
E-mail address: [email protected] (A. Yan). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.03.034
grating [3] (or called as crossed-beam volume grating) is often encountered in many practical applications. The size of the volume grating in the 2-D plane in which the grating vector lies is restricted. They can easily form right-angle diffraction grating in waveguide devices and support the most space-efficient device designs [4,5]. As we all know, the most commonly used theory for diffraction by volume holograms is the coupled-wave theory of Kogelnik [6]. It can be also applied to deal with the problem of anisotropic diffraction [1,2,7,8] in photorefractive crystals. Recently, Montemezzani and Zgonik [9] modified the coupled-wave theory of Kogelnik to include the effects of mixed phase and absorption for grating vector and medium oriented in arbitrary directions to the case of anisotropic materials. The resultant widely used simple analytical expressions
ARTICLE IN PRESS A. Yan et al. / Optik 118 (2007) 418–424
2. Experiment setup and results A schematic diagram of experimental arrangement for observing the anisotropic Bragg diffraction is shown in Fig. 1. The sample used in our experiments is a single domain (0.15 wt% Fe2O3) and oxidized Fe: LiNbO3 crystals with 4 4 10 mm3. A He–Ne laser beam (l ¼ 632:8 nm), which is polarized as an extraordinary light using a half-wave plate is split into two beams with nearly equal intensity 10.5 mW. We expand the two beam sizes, whose diameters are nearly both 4 mm. The two beams are incident upon the two different surface of
BE
M
M BS
1 λ 2
C-axis(z)
.
He-Ne Laser
for isotropic [6] and anisotropic [9] gratings have been found to yield extremely good results when the corresponding assumptions are satisfied. However, these analyses are for the diffraction of infinite and planar boundary volume gratings rather than the practical case of finite-sized gratings. Previous studies [10–12] of the diffraction for finite-sized volume holographic gratings are mostly discussed their isotropic diffraction properties. Solymar [13] has treated such volume gratings with a fully 2-D coupled-wave theory. And analytical solutions with an elegant form have been obtained for certain geometrical configurations. But in this theory, the polarization states of incident and diffracted beam are identical; no investigations of anisotropic diffraction are treated. Generally, with the rapid development of the minimization and integration of optical elements in integrated optics and optical communication, one frequently encounters anisotropic diffraction problem with the transformation of polarization states for finite-sized volume gratings. Therefore, a knowledge of the anisotropic diffraction characteristics of such gratings would be very valuable in practical applications. The present investigation concentrates on the problem of anisotropic diffraction for the most interesting case of finite-sized volume holographic gratings recorded in photorefractive crystal. As for the specific recorded isotropic holograms, when the anisotropic Bragg condition is satisfied, the readout beam can make the anisotropic diffraction beam appear. Corresponding experimental results are presented in Section 2. The detailed phase-matching condition of anisotropic diffraction is discussed in the following Section 3. In Section 4, consistent with the 2-D coupled-wave theory, an analytical solution of the diffraction beam has been obtained. Furthermore, we present a detailed theoretical analysis of the several diffraction properties such as angular and wavelength selectivity, trade off between high efficiency and reconstruction fidelity of finite-sized photorefractive volume holographic gratings. A conclusion is given in Section 5.
419
SH
Crystal CCD
M
Fig. 1. Experimental setup: M, rotatable mirror; 1/2l, halfwave plate; BE, beam expander; BS, beam splitter; SH, shutter.
Fig. 2. Experimental results: the photos of diffracted images of (a) isotropic and (b) anisotropic diffraction observed on a screen behind the crystal.
4 mm 4 mm plane with the intersection angle of 2y0 r outside crystal (y0r ¼ 451). Optical axis c is perpendicular to the plane containing the grating vector and the wave vector of the readout beam. We assume that the propagation and polarization directions of the incident beam are in the (x,y) plane, i.e., the ordinary mode. And the wave polarized along z direction is defined as extraordinary mode. A reflection mirror mounted on a fine rotating stage is used in readout phase for rotating the incidence angle of the readout beam. A CCD detector with viewing screen is placed behind the crystal and is parallel to the output surface of the diffracted beams. We used the two beam interference scheme to record the grating that is a similarity of usually used in holographic experiments. The only difference is the two recording beams entering two different crystal surfaces. In the experiment, the intersection angle inside the crystal is equal to 901 because the two recording beams are incident normal to the crystal surfaces, respectively. After the recording, one of the extraordinary beams is blocked and the other extraordinary beam is diffracted from the written grating. Therefore, the isotropic diffraction (or called as EE diffraction) can take place because of satisfying automatically the
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isotropic Bragg condition. A diffracted image can be observed on the screen and is shown as in Fig. 2(a). Then we changed the readout angle at about the Bragg readout angle y0 i by rotating the reflection mirror. That was chosen to read the recorded grating with the angle y0 i to satisfy the Bragg matching condition of anisotropic diffraction which will be discussed in next section in detail. We can observe the anisotropic diffraction (or called as extraordinary to ordinary (EO) diffraction) at the Bragg diffraction angle y0 d using the CCD detector. Photograph of the anisotropic diffracted image is shown in Fig. 2(b).
where e0 and De are the unperturbed and perturbed permittivity tensor of the photorefractive crystal, no and ne are the ordinary and extraordinary indices, respectively; the reduced subscript notation rij represents rijk in accordance with the widely used convention. Now, for the symmetrical recording geometry of the photoinduced refractive index grating as shown in Fig. 3, the grating vector Kg is along y axis and the two off-diagonal elements n2o n2e r42 E sc are nonzero. y Therefore, in this case the EO diffraction of an extraordinary wave into an ordinary wave and vice versa is possible. The phase-matching condition required for observing anisotropic diffraction is
3. Anisotropic Bragg matching condition in LiNbO3 crystal
K g ¼ k i kd
Diffraction of volume grating is a process of wave coupling. Generally, the generation of anisotropic Bragg diffraction needs two necessary conditions, i.e., the interacting beams are phase matched and the dielectric tensor elements involved are nonzero. Photorefractive ferroelectric LiNbO3 is a typical electro-optic crystal. In crystal inhomogeneous illumination excites electrons from defects into the conduction band. These free charge carriers migrate and trapped in darker regions, and space-charge fields build up. For simplicity, we assume that the two recording beams are both extraordinary polarization and the space-charge field is steady-state and written as E ¼ E 0 ½expðjK g rÞ þ expðjK g rÞK^ g , sc
where E0 is the amplitude of the space-charge field, K^ g ¼ K g =jK g j is unit grating vector. This space-charge field modulates the index of refraction through the linear electro-optic effect [14], 1 D ¼ rjkl E sc (1) l , jk where e is the relative permittivity tensor, D(1/e)jk is the change in the jkth element of 1/e, and rjk is the electrooptic tensor. For a LiNbO3 crystal (of 3 m point group) of which the optical axis is parallel to z axis, the permittivity tensor that is subject to the electro-optic effect with the photoinduced space charge fieldE sc j ¼ sc sc ½E sc ; E ; E , can be expressed as, x y z
2
n2o 6 ¼ 0 þ D ¼ 4 0 0
0 n2o 0
(3)
and also illustrated by the readout wave-vector diagram shown in Fig. 4. In other words, the triangle consisting of incident light wave vector ki, diffracted light wave vector kd and grating vector Kg should be complete. Based on momentum conservation principle, we can write the Bragg condition for the anisotropic diffraction in (x, y) plane of uniaxial crystal [15] as following expressions, ne sin yi þ no sin yd ¼ 2ne sin yr , ne cos yi ¼ no cos yd ,
ð4Þ
sinð45 y0i Þ ¼ ne sinð45 yi Þ, sinð45 y0d Þ ¼ no sinð45 yd Þ, sinð45 y0r Þ ¼ ne sinð45 yr Þ,
ð5Þ
where yr, yi and yd are the internal writing angles of the two incident beams and one diffracted beam with respect to the x-axis, y0 r, y0 i and y0 d are the corresponding external writing angles and diffraction angle, respectively. For a grating written with 632.8 nm at an incident angle of 451, Eqs. (4) and (5) theoretically predict a Bragg angle of 40.071 and 50.511 for incidence and diffraction y0 i and y0 d of anisotropic diffraction. (principal refractive indices of LiNbO3 crystal at l ¼ 632:8 nm are: no ¼ 2:286, ne ¼ 2:200)[16]. So the variation of the readout angles for isotropic and anisotropic diffraction in theory can be calculated as dy0 ¼ 4:931.
3 3 2 n4 ðr E sc r E sc Þ n4o r22 E sc n2o n2e r42 E sc 13 z 0 o 22 y x x 6 sc sc 7 2 2 n4o r22 E sc n2o ðr22 E sc 07 5þ6 x y þ r13 E z Þ no ne r42 E y 7 4 5 sc sc 2 2 4 n2e n2o n2e r42 E sc n n r E n r E 42 33 o e e x y z
(2)
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421
y y Grating boundary crystal
r
0
Kg x
i
r 45°
d
R
S
i
WS
i
d 45°
x
d
WR (a)
(b)
Fig. 3. (a) Geometry for the recording and anisotropic diffraction of the finite-sized volume grating and (b) the refractive at crystal surface.
denoted as ki, kd and polarization unit vectors ei, ed. Here ki ¼ ki ðx cos yi þ y sin yi Þ, ki ¼ 2pne =l. From Eq. (3), we can obtain the wave vector of diffracted beam kd ¼ kd ðx cos yd y sin yd Þ, where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kd ¼ k2i þ K 2g 2ki K g sin yi , sin yd ¼ ðK g ki sin yi Þ
x
Kg
ki
i
d
=kd and cos yd ¼ ðki cos yi Þ=kd . To determine the wave coupling, the total optical field Eq. (6) must also approximately satisfy the wave equation:
kd
y
r2 EðrÞ þ o2 m0 ½0 þ DEðrÞ ¼ 0,
Fig. 4. Wave-vector diagram illustrating how anisotropic diffraction is produced via a finite-sized grating of wave vector Kg in LiNbO3 crystal (optical axis c is perpendicular to the incidence plane).
4. Anisotropic diffraction in finite-sized volume grating We shall investigate in this section the anisotropic Bragg diffraction of a finite-sized photorefractive volume grating recorded by two finite plane waves of uniform amplitude distribution. When such a monochromatic optical beam, entering the photorefractive crystal at or near the Bragg angle, is diffracted by an existing photoinduced refractive index grating, only two significant light beams are assumed to be present, namely, the diffracted and transmitted beams. The total optical field in the grating may be written in phasor form as EðrÞ ¼ ei R0 ðrÞ expðjki rÞ þ ed S0 ðrÞ expðjkd rÞ,
where o is the angular frequency of the light inside the photorefractive crystal. For the EO type anisotropic diffraction mode (the propagation and polarization directions of the diffracted wave are in the xy plane and the incident wave polarized along z direction), we can rewrite the total optical field vector as 2 3 edx S0 ðx; yÞ exp½jkd ðx cos yd y sin yd Þ 6 7 EðrÞ ¼ 4 edy S 0 ðx; yÞ exp½jkd ðx cos yd y sin yd Þ 5. eiz R0 ðx; yÞ exp½jki ðx cos yi þ y sin yi Þ (8) Substituting Eqs. (2) and (8) into the wave equation, with the slowly varying envelope assumption (SVEA) and after some algebra, using the relations e2dx þ e2dy ¼ 1 and eiz ¼ 1, we can derive the following coupled-wave equations for the readout and diffracted beams, cos yi
qR0 qR0 þ sin yi ¼ jkR S 0 , qx qy
(9a)
cos yd
qS 0 qS 0 sin yd þ jWS0 ¼ jkS R0 , qx qy
(9b)
(6)
where R0(r) and S0(r) are the complex amplitudes of the readout and diffracted beams, with propagation vectors
(7)
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where kR ¼ ðo2 =2c2 ki Þn2o n2e r42 E 0 ðed K^ g Þ, kS ¼ ðo2 =2c2 kd Þn2o n2e r42 E 0 ðed K^ g Þ and W denotes the deviations form the Bragg condition W ¼ ððo2 =c2 Þn2o k2d Þ=2kd . As shown by Parry and Solymar [17], introducing a new coordinate system (l, m) defined by l ¼ x sin yi y cos yi and m ¼ x sin yd þ y cos yd results in a simplified version of the coupled wave equations, with R0(x, y) and S0(x, y) being transformed into R(x, y) and S(x, y). The directions of the coordinates l and m are at right angles to the directions of the reference and signal beams in readout phase, respectively. The new equations are q2 S þ k0R k0S S ¼ 0 qlqm
and
1 qS , R¼ 0 jkS ql
(10)
where Rðl; mÞ ¼ R0 ðl; mÞ expðjW0 lÞ, Sðl; mÞ ¼ S0 ðl; mÞ expðjW0 lÞ, k0R ¼ kR = sinðyi þ yd Þ, k0S ¼ kS = sinðyi þ yd Þ and W0 ¼ W= sinðyi þ yd Þ. The boundary conditions may be expressed in the l and m coordinate system as S 0 ðl; f ðlÞÞ ¼ 0 on m ¼ f ðlÞ,
(11)
qS 0 ¼ jk0S expðjW0 lÞ on m ¼ qðlÞ, (12) ql where f ðlÞ ¼ sinðyd þ yr Þ= sinðyi yr Þl and qðlÞ ¼ sinðyd yr Þ= sinðyi þ yr Þl. From Fig. 3, we can see that the output boundaries of the transmitted and diffracted beams are parallel to the input boundaries, respectively. But the curves of the output boundaries are very complex and not easy to express the amplitude distributions of the two output beams. So we introduce another coordinate system u ¼ x sin yr y cos yr and v ¼ x sin yr þ y cos yr . In the other new coordinate system, the boundary conditions of output beams can be simplified as u ¼ W R and v ¼ W S . The relationship between (l, m) and (u, v) coordinate systems is written as l ¼ gðu; vÞ ¼ sinðyi þ yr Þu þ sinðyi yr Þv= sin 2yr , u sinðyd yr Þ þ v sinðyd þ yr Þ m ¼ hðu; vÞ ¼ . sin 2yr
The diffraction efficiency [19] is defined as DE ¼ Pd =Pi , 1
R
(16) 0
ÞS 0 ðW R ; v0 Þ dv0
and Pi ¼ where S0 ðW R ; v R Pd ¼ Z Z 1 R0 ðu0 ; 0ÞR0 ðu0 ; 0Þ du0 are respectively the total energy of the diffracted beam and incident readout beam, Z is the average characteristic impedance of the medium.
5. Numerical results and discussions In this section, we discuss the diffraction properties of the finite-sized volume grating utilizing solutions Eq. (15) of coupled wave equations. Fig. 5 reveals the evolution of the coupling process for various values of space charge field amplitude E0 (0.04, 0.1, 0.8 V/mm,) for an on-Bragg anisotropic diffraction, the corresponding diffraction efficiency is indicated in parentheses. As can be expected, the amplitude of diffracted beam and the diffraction efficiency increase with the value of E0 because of the increasing value of the refractive index modulation depth and the coupling constant. In some practical applications, a high efficiency of finite-sized volume hologram is of course needed. On the other hand, from Fig. 5 we can show that higher diffraction efficiencies imply gross distortion of the reconstructed amplitude distribution for the discussed uniform modulation grating. The recording and readout beams we assumed are plane, the amplitude distributions uniform and have a finite lateral width. But the diffraction amplitude deviates more and more from the uniform amplitude distribution. It may be concluded that there is a trade-off between high efficiency and reconstructed uniform beam-profile. This characteristic is very similar to that of isotropic diffraction finite-sized volume grating. If we want to obtain simultaneously with both
(13)
2.0 E0=0.8 V/µm (62%) E0=0.1 V/µm (39%)
(14)
E0=0.04 V/µm (6%)
1.5
The solution for the diffracted wave of the secondorder hyperbolic partial differential equation with its corresponding boundary conditions may be obtained by means of the so-called Riemann method [18]. Then we have Z lA S 0 ðwR ; vÞ ¼ jk0S exp½jW0 gðW R ; vÞ expðiW0 l 0 Þ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J 0 ð2 k0R k0S ½l 0 gðW R ; vÞ½f ðl 0 Þ hðW R ; vÞÞ dl 0 ,
ð15Þ where J0 and J1 are, respectively Bessel functions of the zeroth and first order, l A ¼ W R sinðyi þ yr Þ= sin 2yr .
|S|
422
1.0
0.5
0.0
0
1
2 v (mm)
3
4
Fig. 5. Amplitude profiles of diffracted beam for several values of space charge field E0, using W R ¼ W S ¼ 4 mm.
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high diffraction efficiency and good reconstruction fidelity, some special techniques for optimizing beam uniformity such as quasi-plane wave illumination, nonuniform grating strength modulation, ideal boundary shape, et al. must be used. This phenomenon can also be of great practical importance in the optimizing design of a VHOE device. In many practical cases, it is of interest to investigate the anisotropic diffraction behavior at or near the Bragg matching condition. To fulfill the above condition, we introduce the Bragg mismatching parameter in coupled wave equation Eq. (9b) and which is defined by 2 o 2 pðn2o n2e Þ ne 2 W¼ þ Kg n k 2kd ¼ d o 2 no l c no sin yi
K 2g 4pno
l.
ð17Þ
In this section, we assume that when the Bragg condition Eq. (3) is satisfied, the Bragg incident angle and corresponding wavelength are, respectively denoted as yi0 and l0. The parameters of Dy ¼ yi yi0 and Dl ¼ l l0 are the angle and wavelength deviations from the Bragg condition, respectively. And the deviations Dy and Dl are small. Therefore, a first-order Taylor series expansion of Eq. (16) yields the following expression: " # K 2g ne pðn2o n2e Þ Dl W ¼ K g cos yi0 Dy Dl þ . no l0 l0 no 4pno (18) Note Eq. (18) differs from the corresponding expression for W in Ref. [6]. Figs. 6(a and b) show the dependence of the diffraction efficiency DE on the angular and wavelength mismatch parameters Dy and Dl (W R ¼ W S ¼ 4 mm, l0 ¼ 633 nm, y0 ¼ 42:761, E 0 ¼ 0:04 V=mm). It is demonstrated clearly that the diffraction efficiency of anisotropic diffraction decreases dramatically for small values of Dy in Fig. 6(a). Its maximum value concentrates at the center coordinate, whereas Dy is zero. The same situation also can be found for Dl in Fig. 6(b). These effects are due to the loss of coupling between the two beams inside the volume grating when off-Bragg parameters are introduced. It can be concluded that finite-sized anisotropic volume gratings exhibit strong angular and wavelength selectivity. The phenomena can find applications in the fields of optoelectronics such as optical data storage, angular and wavelength filters in DWDM systems.
6. Conclusions The anisotropic Bragg diffraction of light by finitesized volume grating in photorefractive crystals is
Fig. 6. (a) The variation curve of diffraction efficiency as a function of angular deviation Dy, Dl ¼ 0. (b) The variation curve of diffraction efficiency as a function of wavelength deviation Dl, Dy ¼ 0.
investigated in this paper. This diffraction can take place when the Bragg phase matching condition is satisfied and the involved off-diagonal tensor elements are nonzero. We also have identified the anisotropic diffraction behavior in experiment for the volume grating recorded in Fe:LiNbO3 crystal; the photos of diffracted images of isotropic and anisotropic diffraction are given. The theoretical model for the anisotropic diffraction of finite-sized volume grating is presented at or near Bragg diffraction. The model is based on the two-wave first-order coupled wave theory framework developed by Solymar. Analytical solutions are derived for amplitudes of the transmitted and diffracted beams. Numerical calculations show that, for EO-type Bragg diffraction of uniform plane waves, high reconstruction fidelity is often accompanied by a decrease in diffraction efficiency. The phenomenon can be of an important problem considered in practical design of VHOE. For deviations from the anisotropic Bragg condition, it is
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shown that the diffraction efficiency depends strongly on the deviation parameter Dy and Dl. In particularly, the finite-sized anisotropic volume grating exhibits high angular and wavelength selectivity. All the results are useful for the design of volume holographic optical elements based on finite-sized anisotropic volume grating.
Acknowledgments The authors acknowledge the supports of the Science and Technology Committee of China (Grant No. 2002CCA03500) and the National Nature Science Foundation of China (Grant No. 60177016).
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